2 research outputs found
Fooling sets and rank
An matrix is called a \textit{fooling-set matrix of size }
if its diagonal entries are nonzero and for every
. Dietzfelbinger, Hromkovi{\v{c}}, and Schnitger (1996) showed that
n \le (\mbox{rk} M)^2, regardless of over which field the rank is computed,
and asked whether the exponent on \mbox{rk} M can be improved.
We settle this question. In characteristic zero, we construct an infinite
family of rational fooling-set matrices with size n = \binom{\mbox{rk}
M+1}{2}. In nonzero characteristic, we construct an infinite family of
matrices with n= (1+o(1))(\mbox{rk} M)^2.Comment: 10 pages. Now resolves the open problem also in characteristic